Problem: Jesse tossed a paint brush off her roof. The height of the brush (in meters above the ground) $t$ seconds after Jesse tossed it is modeled by $h(t)=-5t^2+5t+10$ Jesse wants to know when the brush will hit the ground. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. $h(t)=$ 2) How many seconds after being thrown does the brush hit the ground?
Solution: Choosing a form When the brush hits the ground, its height above the ground is $0$ meters. So we're looking for what values of $t$ make the output of the function $0$. Which form reveals this feature? Here's a summary of what each form reveals along with examples. Note that these are all equivalent forms of the same function, but not the function modeling the height of the brush. Form Example Feature revealed Standard $f(x)=2x^2-12x+{10}$ $y$ -intercept is ${10}$ Factored $f(x)=2(x-C{1})(x-C{5})$ Zeros are $x=C1$ and $x=C5$ Vertex $f(x)=2(x-{3})^2{-8}$ Vertex is $(3,{-8})$ Rewrite in factored form The zeros of the function tell us which values of $t$ make the output of the function $0$, so let's rewrite $h(t)$ in factored form: $\begin{aligned} h(t)&=-5t^2+5t+10 \\\\ &=-5(t^2-t-2) \\\\ &=-5\left(t+1\right)\left(t-2\right) \end{aligned}$ When does the brush hit the ground? The factored form of the function reveals its zeros: $\begin{aligned} &0=-5(t+1)(t-2) \\\\ &t+1=0\text{ or }t-2=0 \\\\ &\xcancel{t=-1} \text{ or }t=2 \end{aligned}$ A negative value for time doesn't make sense in this context, so the brush hits the ground at $t=2$ seconds. Answers 1) The factored form of the function reveals when the brush hits the ground: $h(t)=-5\left(t+1\right)\left(t-2\right)$ 2) The brush hits the ground $2$ seconds after being thrown.